Sesquitertium
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Sesquitertium refers to the improper vulgar fraction
- <math> {4 \over 3} = 1 + {1 \over 3}. <math>
It is a superparticular number.
The sesquitertium in musical harmony is the ratio corresponding to the interval diatessaron.
In visual harmony, the sesquitertium can be the ratio between the length and the width of a rectangle. For example, a 16 inch CRT monitor whose length is 13 inches and whose width is 9.5 inches has a length-to-width ratio of 1.368 which is close to a sesquitertium, 1.333, with an error of +2%.
See also: sesquialterum, sesquiquartum, sesquiquintum.
External link
- De Institutione Arithmetica, liber II (http://www.music.indiana.edu/tml/6th-8th/BOEARI2_TEXT.html) by Anicius Manlius Severinus Boethius. (Text in Latin)
- Idem contingit etiam in triplicibus. The same happens also in triples (multiples of three).
- Ex illis enim sesquitertii procreantur. Out of them [triples], sesquitertii can truly be begotten.
- Nam quoniam primus triplex est ternarius numerus, habet unum sesquitertium, id est .IIII., cuius quaternarii tertia pars non potest inveniri. For because the first triple is the number three, it has one sesquitertium, it is 4, the fourth [number] of which its third part cannot be found.
- Secundus vero, qui est .VIIII., habet ad se duodenarium numerum sesquitertium, duodenarius autem, quoniam habet tertiam partem, in sesquitertia proportione comparatur ad eum numerus .XVI., qui tertiae partis sectione solutus est. The second [case] verily, which is 9, has next to itself the number 12 as its sesquitertium, 12 however, since it has a third part, the number 16 is coupled to it in sesquitertial proportion, [16] which by cutting has been freed of any third part.
- .XXVII., autem, quoniam tertius est triplex, habet ad se sesquitertium .XXXVI. et hic rursus ad .XLVIII. eadem proportione comparatur. 27, however, since — thirdly — it is a triple, has next to itself the sesquitertium 36 and this [36] in return [reaches next] to 48, the two of which in the same proportion [sesquitertium] are being coupled.